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1. Natural transformations Let C and D be categories, and F, G : C → D functors. A natural transformation T : F → G is a collection of morphisms T (x) : F (x) → G(x), one for each object x of C, such that for each morphism f : x → y in C, the diagram F (x) T (x) F (f ) F (y) / G(x) G(f ) T (y) / G(y) commutes. If S : F → G is a natural transformation and T : G → H is another, then T ◦ S : F → H is a natural transformation, defined by (T ◦ S)(x) = T (x) ◦ S(x). There is also an identity natural transformation F → F for any functor. A natural isomorphism T : F → G is a natural transformation which has an inverse S : G → F . Equivalently, it is a natural transformation such that for each object x of C, the morphism T (x) : F (x) → G(x) in D is an isomorphism. Example 1.1. The homomorphism ∂∗ : Hn (X, A) → Hn−1 (A) is a natural transformation. It is not a natural isomorphism. Example 1.2. In Bredon IV.3 we defined for each pointed topological space (X, x0 ) a homomorphism of abelian groups π1 (X, x0 )/[π1 , π1 ] → H1 (X). This gives a natural transformation between two functors from the category of pointed topological spaces to the category of abelian groups. If we restrict to the subcategory of path connected pointed spaces, it becomes a natural isomorphism. 2. Representable functors and Yoneda’s lemma Let Set be the category of sets, and let C be an arbitrary category. For each object x of C, we get a functor MorC (x, −) : C → Set. This is called the (covariant) functor represented by x. An arbitrary (covariant) functor F : C → Set is called representable if it is naturally isomorphic to MorC (x, −) for some x. Let C be a category and F : C → Set a functor. Let x ∈ C, and let MorC (x, −) : C → Set be the functor represented by x. If T : MorC (x, −) → F is a natural transformation, it in particular gives a map T (x) : MorC (x, x) → F (x), which we can evaluate on the identity map Idx and get an element T (x)(Idx ) ∈ F (x). Yoneda’s lemma is the following statement. Lemma 2.1. The association T 7→ T (x)(Idx ) gives a bijection Nat(MorC (x, −), F ) → F (x) from the set of all natural transformations to the set F (x). Proof sketch. Given c ∈ F (x), we can get a natural transformation Tc : MorC (x, −) → F by Tc (y) MorC (x, y) −−−→ F (y) f 7→ F (f )(c) This process is inverse to the process T 7→ T (x)(Idx ).